Subgroups — #15768169

Order 3328.

1646 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718873	C₂	1	2	yes	maximal
2	C₂×#11718143	C₂	1	4	yes	maximal, complemented
3	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	—	2	4	no	—
6	C₂³×C₈×C₁₃	—	2	8	no	—
7	C₂³×C₈×C₁₃	C₄	1	4	yes	complemented
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₂²⋊Dic₁₃	—	2	2	no	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
11	#11718873	—	2	4	no	—
12	#11718873	C₄	1	8	yes	complemented
13	C₂×#88071	—	2	2	no	—
14	#11718143	—	2	8	no	—
15	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	—	2	4	no	—
17	C₂³×C₄×C₁₃	—	2	4	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	D₄	1	4	yes	—
22	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂³×Dic₁₃	—	2	4	no	—
25	C₂²×C₈×C₁₃	—	2	32	no	—
26	C₂²×C₈×C₁₃	—	2	8	no	—
27	C₂²×C₈×C₁₃	—	2	4	no	—
28	C₂²×C₈×C₁₃	—	2	4	no	—
29	C₂²×C₈×C₁₃	—	2	16	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	D₄	1	8	yes	—
32	C₂×C₂²⋊Dic₁₃	—	2	8	no	—
33	C₂×C₂²⋊Dic₁₃	—	4	4	no	—
34	C₂×C₂²⋊Dic₁₃	—	4	4	no	—
35	C₂×C₄⋊Dic₁₃	—	2	8	no	—
36	C₂×C₄⋊Dic₁₃	—	2	8	no	—
37	C₂×C₄⋊Dic₁₃	—	2	4	no	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
39	#88071	—	2	8	no	—
40	#10721	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	—	2	4	no	—
42	C₂⁴×C₁₃	—	2	4	no	—
43	C₂⁴×C₁₃	C₂²⋊C₄	1	4	yes	—
44	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂⁴×C₁₃	D₈	1	2	yes	—
46	C₂⁴×C₁₃	SD₁₆	1	4	yes	—
47	C₂⁴×C₁₃	Q₁₆	1	2	yes	—
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	—	2	16	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	4	yes	—
67	C₂²×Dic₁₃	—	2	8	no	—
68	C₂²×Dic₁₃	—	4	8	no	—
69	C₂²×Dic₁₃	—	4	4	no	—
70	C₂²×Dic₁₃	—	4	4	no	—
71	C₂²×Dic₁₃	—	4	8	no	—
72	C₂×C₈×C₁₃	—	2	32	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	16	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	complemented
77	C₂²⋊Dic₁₃	—	4	16	no	—
78	C₄⋊Dic₁₃	—	2	16	no	—
79	C₄⋊Dic₁₃	—	4	4	no	—
80	C₄⋊Dic₁₃	—	4	4	no	—
81	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
82	C₂×#380	—	13	2	no	—
83	C₂×#340	—	13	4	no	—
84	C₂³×C₁₃	—	2	4	no	—
85	C₂³×C₁₃	—	2	8	no	—
86	C₂³×C₁₃	—	2	8	no	—
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	—	2	16	no	—
89	C₂³×C₁₃	—	2	8	no	—
90	C₂³×C₁₃	—	2	8	no	—
91	C₂³×C₁₃	#95	1	4	yes	—
92	C₂³×C₁₃	#102	1	8	yes	—
93	C₂³×C₁₃	#103	1	8	yes	—
94	C₂³×C₁₃	#106	1	4	yes	—
95	C₂³×C₁₃	#107	1	4	yes	—
96	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₃	#121	1	4	yes	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
101	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂³×C₁₃	#135	1	4	yes	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	8	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	16	no	—
109	C₂×C₄×C₁₃	#95	1	4	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
116	C₂×C₄×C₁₃	#120	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	#123	1	4	yes	—
119	C₂×Dic₁₃	—	4	8	no	—
120	C₂×Dic₁₃	—	4	8	no	—
121	C₂×Dic₁₃	—	4	8	no	—
122	C₂×Dic₁₃	—	8	8	no	—
123	C₂×Dic₁₃	—	8	8	no	—
124	C₈×C₁₃	—	2	8	no	—
125	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
126	C₂³×C₈	—	2	4	no	—
127	C₂³×C₈	—	2	8	no	—
128	C₂³×C₈	Dic₁₃	1	4	yes	complemented
129	C₂³×C₈	C₂×D₁₃	1	2	yes	—
130	C₂²×C₄⋊C₄	—	13	4	no	—
131	C₂²×C₂²⋊C₄	—	26	2	no	—
132	#380	—	13	8	no	—
133	#380	—	26	4	no	—
134	C₂×#95	—	26	2	no	—
135	#340	—	26	8	no	—
136	C₂²×C₁₃	—	2	8	no	—
137	C₂²×C₁₃	—	2	8	no	—
138	C₂²×C₁₃	—	2	16	no	—
139	C₂²×C₁₃	—	2	16	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	—	2	4	no	—
142	C₂²×C₁₃	#340	1	8	yes	complemented
143	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
144	C₂²×C₁₃	#380	1	2	yes	—
145	C₂²×C₁₃	#386	1	4	yes	—
146	C₂²×C₁₃	C₂×#102	1	2	yes	—
147	C₂²×C₁₃	C₂×#103	1	2	yes	—
148	C₂²×C₁₃	#416	1	4	yes	—
149	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
151	C₂²×C₁₃	#427	1	2	yes	—
152	C₂²×C₁₃	#465	1	8	yes	—
153	C₂²×C₁₃	#466	1	4	yes	—
154	C₂²×C₁₃	#467	1	4	yes	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
157	C₄×C₁₃	#379	1	1	yes	characteristic
158	C₄×C₁₃	#381	1	1	yes	characteristic
159	C₄×C₁₃	#385	1	4	yes	—
160	Dic₁₃	—	8	8	no	—
161	C₂⁵	D₅₂	1	1	yes	characteristic
162	C₂³×C₄	—	2	4	no	—
163	C₂³×C₄	—	2	4	no	—
164	C₂³×C₄	—	26	4	no	—
165	C₂³×C₄	Dic₂₆	1	2	yes	—
166	C₂³×C₄	C₄×D₁₃	1	4	yes	—
167	C₂³×C₄	D₅₂	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
169	C₂³×C₄	C₂²⋊D₁₃	1	4	yes	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²×C₈	—	2	32	no	—
172	C₂²×C₈	—	2	8	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	16	no	—
176	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
177	C₂²×C₈	C₂²⋊D₁₃	1	8	yes	—
178	C₂×C₄⋊C₄	—	13	8	no	—
179	C₂×C₄⋊C₄	—	26	8	no	—
180	C₂×C₄⋊C₄	—	26	8	no	—
181	C₂×C₄⋊C₄	—	26	4	no	—
182	C₂×C₂²⋊C₄	—	26	8	no	—
183	C₂×C₂²⋊C₄	—	52	4	no	—
184	C₂×C₂²⋊C₄	—	52	4	no	—
185	#95	—	26	8	no	—
186	C₂×C₁₃	—	2	8	no	—
187	C₂×C₁₃	C₂²⋊#95	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
189	C₂×C₁₃	#1735	1	1	yes	characteristic
190	C₂×C₁₃	#1807	1	2	yes	—
191	C₂×C₁₃	#1808	1	2	yes	—
192	C₂×C₁₃	#1892	1	4	yes	—
193	C₂×C₁₃	#1893	1	4	yes	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	#6095	1	4	yes	—
197	C₂⁴	D₁₀₄	1	2	yes	—
198	C₂⁴	Dic₅₂	1	2	yes	—
199	C₂⁴	C₄⋊Dic₁₃	1	2	yes	—
200	C₂⁴	#6103	1	4	yes	—
201	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	26	8	no	—
210	C₂²×C₄	—	52	4	no	—
211	C₂²×C₄	—	52	4	no	—
212	C₂²×C₄	—	52	8	no	—
213	C₂²×C₄	—	52	8	no	—
214	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
215	C₂²×C₄	#6101	1	8	yes	—
216	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
217	C₂²×C₄	#6103	1	4	yes	—
218	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
219	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
221	C₂²×C₄	#6127	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
225	C₂×C₈	—	2	32	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	16	no	—
229	C₂×C₈	C₂²⋊Dic₁₃	1	8	yes	complemented
230	C₄⋊C₄	—	26	16	no	—
231	C₄⋊C₄	—	52	4	no	—
232	C₄⋊C₄	—	52	4	no	—
233	C₂²⋊C₄	—	52	16	no	—
234	C₁₃	#10721	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	16	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	#88056	1	8	yes	—
243	C₂³	#88057	1	4	yes	—
244	C₂³	#88058	1	4	yes	—
245	C₂³	#88061	1	8	yes	—
246	C₂³	#88071	1	4	yes	—
247	C₂³	C₂×#6095	1	2	yes	—
248	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
249	C₂³	#88158	1	4	yes	—
250	C₂³	C₂×Dic₅₂	1	1	yes	characteristic
251	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
252	C₂³	C₂×#6103	1	2	yes	—
253	C₂³	#88184	1	4	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	52	8	no	—
261	C₂×C₄	—	52	8	no	—
262	C₂×C₄	—	52	8	no	—
263	C₂×C₄	—	104	8	no	—
264	C₂×C₄	—	104	8	no	—
265	C₂×C₄	#88071	1	4	yes	—
266	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#6101	1	2	yes	—
268	C₂×C₄	#88178	1	4	yes	—
269	C₂×C₄	#88180	1	2	yes	—
270	C₂×C₄	C₄×C₂²⋊D₁₃	1	8	yes	—
271	C₂×C₄	#88183	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
274	C₂×C₄	#88207	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₂²	—	2	16	no	—
277	C₂²	—	2	16	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	8	no	—
282	C₂²	#11718143	1	8	yes	complemented
283	C₂²	C₂×#88056	1	2	yes	—
284	C₂²	C₂×#88057	1	1	yes	characteristic
285	C₂²	C₂×#88058	1	1	yes	characteristic
286	C₂²	#11718767	1	2	yes	—
287	C₂²	C₂×#88061	1	2	yes	—
288	C₂²	#11718774	1	4	yes	—
289	C₂²	#11718775	1	8	yes	—
290	C₂²	#11718776	1	4	yes	—
291	C₂²	#11718777	1	4	yes	—
292	C₂²	C₂×#88071	1	1	yes	characteristic
293	C₂²	#11718873	1	2	yes	—
294	C₂²	#11718874	1	4	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	104	8	no	—
297	C₄	C₄×C₂²⋊Dic₁₃	1	2	yes	—
298	C₄	#11718871	1	4	yes	—
299	C₄	#11718872	1	1	yes	characteristic
300	C₄	#11718908	1	1	yes	characteristic
301	C₂	—	2	8	no	—
302	C₂	#13113059	1	1	yes	characteristic
303	C₂	C₂×#11718143	1	1	yes	characteristic
304	C₂	#13114553	1	1	yes	characteristic
305	C₂	#13114861	1	2	yes	—
306	C₂	#13114865	1	2	yes	—
307	C₂	#13115259	1	4	yes	—
308	C₂	#13115264	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂⁵×C₁₃	D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	#10721
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#11718143	C₂	semidirect
7. C₂³×C₈×C₁₃	C₄	semidirect
12. #11718873	C₄	semidirect
76. C₂×C₈×C₁₃	C₂²⋊C₄	semidirect
128. C₂³×C₈	Dic₁₃	semidirect
142. C₂²×C₁₃	#340	semidirect
229. C₂×C₈	C₂²⋊Dic₁₃	semidirect
234. C₁₃	#10721	semidirect
282. C₂²	#11718143	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718873	2
C₂×#11718143	2
C₂⁴×C₈×C₁₃	2
#10721	13